Nonlinear Schrödinger equation with Coulomb potential
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916215316807680 |
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| author | Miao, Changxing Zhang, Junyong Zheng, Jiqiang |
| author_facet | Miao, Changxing Zhang, Junyong Zheng, Jiqiang |
| contents | In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i\partial_tu+Δu+\frac{K}{|x|}u=λ|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the long range potential $K|x|^{-1}$ on the existence theory and scattering theory for nonlinear Schrödinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. $K>0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1809_06685 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Nonlinear Schrödinger equation with Coulomb potential Miao, Changxing Zhang, Junyong Zheng, Jiqiang Analysis of PDEs In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i\partial_tu+Δu+\frac{K}{|x|}u=λ|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the long range potential $K|x|^{-1}$ on the existence theory and scattering theory for nonlinear Schrödinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. $K>0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms. |
| title | Nonlinear Schrödinger equation with Coulomb potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1809.06685 |